There a number of theorems in General Relativity that are of great importance for the structure of the theory itself, as well as for the solutions to the field equations. These theorems underpin a lot of the acquired intuition on how gravity should function in different environments, and what the resulting phenomenology should be. In alternative theories of gravity, however, the theorems of General Relativity often fail, allowing new behaviours that would otherwise be impossible.
Here we briefly recap what we consider to be some of the most important theorems of General Relativity. In later sections we will show how these theorems are violated in alternative theories, and discuss the consequences of this.
2.4.1. Lovelock’s theorem
Lovelock’s theorem [831, 832] limits the theories that one can construct from the metric tensor alone. To enunciate this theorem, let us begin by assuming that the metric tensor is the only field involved in the gravitational action. If the action can be written in terms of the metric tensor gµν alone, then we can write
S = Z
d4xL(gµν). (56)
If this action contains up to second derivatives of gµν, then extremising it with respect
to the metric gives the Euler-Lagrange expression Eµν[L] = dxdρ ∂L ∂gµν,ρ − d dxλ ∂L ∂gµν,ρλ −∂g∂L µν , (57)
and the Euler-Lagrange equation is Eµν(
L) = 0. Lovelock’s theorem can then be stated as the following:
Theorem 2.1. (Lovelock’s Theorem)
The only possible second-order Euler-Lagrange expression obtainable in a four dimen- sional space from a scalar density of the formL = L(gµν) is
Eµν = α√ −g Rµν −12gµνR + λ√−ggµν, (58) 29
whereα and λ are constants, and Rµν andR are the Ricci tensor and scalar curvature,
respectively.
This powerful theorem means that if we try to create any gravitational theory in a four-dimensional Riemannian space from an action principle involving the metric tensor and its derivatives only, then the only field equations that are second order or less are Einstein’s equations and/or a cosmological constant. This does not, however, imply that the Einstein-Hilbert action is the only action constructed from gµν that results in the
Einstein equations. In fact, in four dimensions or less one finds that the most general such action is L = α√−gR − 2λ√−g + βµνρλRαβ µνRαβρλ+ γ√−g R2− 4Rµ νRνµ+ R µν ρλR ρλ µν , where β and γ are also constants. The third and fourth terms in this expression do not, however, contribute to the Euler-Lagrange equations as
EµνhαβρλRγδαβRγδρλ i = 0 (59) Eµνh√ −gR2 − 4Rα βRβα+ R αβ ρλR ρλ αβ i = 0, (60)
where the action of Eµν on any function X is defined as in Eq. (57). The first of these
equations is valid in any number of dimensions, and the second is valid in four dimensions only.
Lovelock’s theorem means that to construct metric theories of gravity with field equa- tions that differ from those of General Relativity we must do one (or more) of the fol- lowing:
• Consider other fields, beyond (or rather than) the metric tensor.
• Accept higher than second derivatives of the metric in the field equations. • Work in a space with dimensionality different from four.
• Give up on either rank (2,0) tensor field equations, symmetry of the field equations under exchange of indices, or divergence-free field equations.
• Give up locality.
The first three of these will be the subject of the next three sections of this report. The fourth option requires giving up on deriving field equations from the metric variation of an action principle, and will not be considered further here.
2.4.2. Birkhoff ’s theorem
Birkhoff’s theorem7 is of great significance for the weak-field limit of General Rela-
tivity. The theorem states [160]
7This theorem is commonly attributed to Birkhoff, although it was already published two years earlier
Theorem 2.2. (Birkhoff’s Theorem)
All spherically symmetric solutions of Einstein’s equations in vacuum must be static and asymptotically flat (in the absence ofΛ).
Strictly speaking, there are very few situations in the real Universe in which Birkhoff’s theorem is of direct applicability: Exact spherical symmetry and true vacuums are rarely, if ever, observed. Nevertheless, Birkhoff’s theorem is very influential in how we under- stand the gravitational field around (approximately) isolated masses. It provides strong support for the relativistic extension of our Newtonian intuition that far from such ob- jects their gravitational influence should become negligible, or, equivalently, space-time should be asymptotically flat8. We can therefore proceed with some confidence in treat-
ing the weak-field limit of General Relativity as a perturbation about Minkowski space. Birkhoff’s theorem also tells us that certain types of gravitational radiation (from a star that pulsates in a spherically symmetric fashion, for example) are not possible.
As we will show below, Birkhoff’s theorem does not hold in many alternative theories of gravity. We therefore have less justification, aside from our own intuition, in treating the weak field limit of these theories as perturbations about Minkowski space. We must instead be more careful, as the space-time we perform our expansion around can have asymptotic curvature, leading to either time or space-dependence of the background (or some combination of the two). What is more, the perturbations themselves may be time- dependent, and their form can be sensitive to the type of asymptotic curvature that the background exhibits. Behaviours such as these are not expected in General Relativity [837].
2.4.3. The no-hair theorems
These theorems are named after the phrase coined by Wheeler that “black holes have no hair”. The first of these theorems was given by Israel and showed that the only static uncharged asymptotically flat black hole solution to Einstein’s equations is the Schwarzschild solution [650]. He later extended this theorem to include charged objects [651], and Carter extended it to black holes with angular momentum [262]. The theorem is therefore often stated today as “the generic final state of gravitational collapse is a Kerr-Newman black hole, fully specified by its mass, angular momentum, and charge ” [1254].
Complementary to the black hole no-hair theorems is the no-hair ‘theorem’ of de Sitter space. The claim here is that in the context of General Relativity with a cosmological constant all expanding universe solutions should evolve towards de Sitter space. This has been shown explicitly by Wald for all Bianchi type models9 [1255].
These theorems play an important role in General Relativity and cosmology. Some progress has been made in extending them to alternative theories of gravity, but there have also been explicit examples of them being violated in particular theories. This will be discussed further in subsequent sections.
8Of course, in a cosmological setting asymptotic regions are never realised as we will eventually come
across the other masses in the Universe.
9Except type-IX universes with large amounts of spatial curvature.